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Ardern posted an Instagram message on Friday thanking "our wonderful midwife Libby" who she said was not only incredible at her job but also made the mac and cheese.

Meanwhile, dozens of reporters waited for the good news after Ms Ardern was taken to Auckland Hospital for the birth and have been hoping to catch a glimpse of the new family.

The trio are expected to briefly address media on their way home before settling in for a stint away from the limelight - but Ms Ardern's office has now confirmed that won't be happening on Friday.

While everyone was well, "if not a bit tired", they would be spending another night in the ward.

Nurses have described the baby as "very alert and one hungry baby".

She was one of 24 children born at the hospital - the country's largest public facility - on Thursday.

Ms Ardern has been given no preferential treatment during her stay in the maternity ward - although a security detail has been on hand - and she was driven to the hospital by Mr Gayford.

Meanwhile, speculation will also likely turn to baby names, with no clues provided in Thursday's announcements.

Asked about which surname would be given, Ms Ardern in January said she expected the baby to be a Gayford.

"I'll probably try sneak in my name somehow," she said at the time.

However, in recent weeks she's said the picking process was going "terribly".

News of the birth sparked headlines across the globe and celebrations in New Zealand on Thursday.

Malcolm Turnbull joined other world leaders, including Canada's Justin Trudeau and British Prime Minister Theresa May in sending their congratulations.

In a recent interview, Ms Ardern said she was bracing for life in a media bubble for her family.

"We know that we are watched, and we do our best just to be true to ourselves. But we know that that will happen to us for parenting as well," she told TVNZ.

"As long as the judgement falls on me more than my child, that's what I hope for the most."

Ms Ardern has become the first elected world leader to take maternity leave and for the next six weeks will only be consulting on major issues while her deputy, Winston Peters, handles day-to-day matters.